Limitations
Some of the limitations of NES hardware and common mapper hardware severely limit the system's capability to perform well in some genres. This could be part of why they took off after 1991 when more powerful hardware became more readily available.
Simulation
Limitations: 8 KiB PRG RAM; no 3D graphics hardware
Some kinds of simulator games, like SimCity, The Sims, Harvest Moon, and Animal Crossing, have large grid-based world maps and tend to need 8 KiB just to store this map, let alone the rest of the world's state. Very few NES cartridge boards provided more than 8 KiB, and most of them were Japan-only.
Games that allow the player to customize his appearance extensively, as in Animal Crossing and The Sims, typically rely on 3D texture mapping so that the changes to the appearance can be seen from all angles. Tony Hawk series for Game Boy Advance soft-rendered a 3D player character in real time, but the GBA CPU is also fast enough to emulate the NES CPU.
Creation
Limitations: 8 KiB PRG RAM, input devices
Games that allow for modification, such as modified game graphics, maps, and scripts, tend to be extremely limited. There's also no alphabetic keyboard or precise pointing device on an NES. A BASIC interpreter and a "shooter maker" were released only in Japan. Videomation is a paint program with an expanded CHR ROM, but its animation is severely limited compared to Mario Paint for Super NES, and it can't save.
Driving
Limitations: No 3D graphics
There are two ways to do a driving game: behind-the-car with a Pole Position style track (as seen in F-1 Race and Rad Racer) and overhead/isometric style (as seen in RC Pro-Am and Micro Machines.) In a behind-the-car view, the car can't turn all the way around. Overhead view severely limits how far ahead one player can see, and the camera generally has to follow one player or the other.
Puzzle
Limitations: 16x16 pixel attribute tiles
A lot of tile-matching games, such as Magic Jewelry, Yoshi's Cookie, Mystic Pillars, Wario's Woods, Palamedes, and one of the Puyo Puyo games, have 16x16 pixel tiles. This limits each player's playfield to about 6x12 cells. Games with smaller tiles typically can't go over three unique colors, as seen in Dr. Mario and Tetris 2. One compromise is seen in LJ65: tiles in both players' playfields are drawn as a dithered combination of orange, green, and blue. It's not as clean as, say, Tetris for Sega Master System, but it gets the job done.
Fighting
Limitations: Overdraw
If both players' graphics are drawn as sprites, characters will have to be no wider than 32 pixels. Otherwise, they'll flicker like crazy when you cross the 8-sprite limit. You can get away with wider jumping poses because it's less likely that both characters will be in the air at once. Shaq-Fu was poor in execution, but its core idea of smaller characters could be made to work well on an NES. You could aim for something like Super Smash Bros.
The other way to do this, allowing big characters like in Street Fighter, is to draw one character as 8x16 pixel sprites and the other as background tiles. This allows up to 64-pixel-wide characters, or slightly smaller counting projectile attacks. But you have to design the game such that only one player faces either direction, so that you can draw the other player with mirrored sprites. That means you can't easily jump behind or roll past the other player and attack from behind unless the frames for facing the other player's back have dedicated cels. In addition, backgrounds will be plain, and you'll probably need an advanced mapper with a scanline timer to perform multiple scroll splits at the status bar, play area, and ground.
Bullet hell shooter
Limitations: Overdraw, CPU speed, 5-line OAM DMA delay
Shooters can be hard, like Recca, but they can't have more than 64 things in the air at once. Some platforms (like GBA and Neo Geo) can rewrite part of the sprite table during rendering to expand the sprite capability, but the NES can't. And even then, more than 8 on a line will flicker so bad that bullets will be invisible half the time.
Party multiplayer
Limitations: Sprite palettes; overdraw; DMC controller interaction
An NES Four Score hub provides four controller ports, one for each of the four palettes. If each player's car, uniform, etc. has the same graphics with a palette swap, there are no palettes left for anything else. Either other things like projectiles and enemies need to be the same color as the player, or they have to be background objects. Four 16x16 or 32x16 pixels will also hit the 8-sprite limit with no room for projectiles unless the game is designed not to encourage them to be at the same vertical position.
Memory accesses during sample playback occasionally cause a bit from the controllers to be skipped. It takes longer to read 16 bits than 8, so not only is it twice as likely to skip a bit, but rereading becomes less practical. So either your music and sound effects during gameplay will have to be purely PCM, or you'll have to use the signature bits to detect deletions from the bit sequence and use the previous frame's keypresses instead.
Music
Limitations: Audio hardware
Unlike the 60-pin Famicom cart edge, the 72-pin NES cart edge lacks a pair of pins for audio hardware on the cartridge. Instead it has ten pins connected directly to the expansion port. A 47 kΩ resistor from one of the cart pins to an audio pin on the expansion port enables audio out of the expansion port as on a Famicom, but Nintendo never released an accessory with this resistor, and top-loading consoles don't even have this port. So NES games usually have to use the 2A03's internal APU channels.
Music games such as Beatmania, Dance Dance Revolution, and Amplitude use CD-quality audio played from a disc. Though the NES APU is capable of competent covers of popular music, a lot of players demand the original studio recordings, or at least an electric guitar that sounds like an electric guitar and chords that don't sound warbly. Witness poor sales of the Game Boy Color versions of Beatmania and DDR.
Apart from DDR: Disney Dancing Museum for the Japanese N64 and games on Nintendo handhelds, most music games have come out on disc-based consoles. A lot of these music games use a custom controller, shaped like a musical instrument or like the Power Pad. Because cartridges were much more expensive to replicate, bundling a controller with your game cartridge was cost prohibitive.
Disc-based games using a standard controller, like Parappa the Rapper and Frequency, came on discs 600 MB to 4 GB in size and thus could afford to store the music tracks as two long samples: one for background music and one for the foreground instrument. Guitar Hero On Tour games for Nintendo DS are 128 MB each after several Moore's law doublings of ROM density. But with a typical NES ROM size of 128 to 512 KiB, a big fat sample like that is out of the question.
